expressed by a New Formula. 5 



the age of about 12 years to the end of life. Both formulae are 

 similar to the formula which has been shown to represent the 

 law, according to temperature, of the elastic force of steam of 

 maximum density, and the formula which has been shown to 

 represent the law of density of saturated steam (Philosophical 

 Magazine, March and July 1865). All four formulas are deduced 

 — from a differential of the form following : — 



d.h S V = *(] + t -y*dt. 



In the two formulae for human life, the quantity P represents 

 survivors at the absolute age (a + 1) out of a unit of population 

 existing at the absolute age (a) measured from one of the two 

 zeros of life. The quantity a is the rate of decrement in a unit 

 of time at the absolute age (a) whence t is measured, on the 

 assumption that the infinitesimal rate of decrement at age a 

 continues constant for the unit of time. The above equation for 

 d . log P, on integration, yields 



co m lo g P=-^{l-(l + £)-"}, 



wherein 



n=\ -1 = 1-302585. 

 k 



The most remarkable difference between the two new formulae 

 for human life, and the two formulae for elastic force and den- 

 sity of saturated steam, will be found to consist in the relative 

 positions of the ideal zeros of the forces of life and steam. In 

 the case of steam the constant a (which indicates distance from 

 the ideal zero) is very great, and marks a position of such zero 

 far beyond the reach of observation, viz. 276° Centigrade below 

 the temperature of melting ice. In the case of the two formulae 

 for human life, both of the ideal zeros are close at hand, and 

 one of the two zeros may be passed by a living person. In the 

 formula for the period of immaturity or childhood, the value of the 

 constant a at birth -time is %\ years very nearly ; that is to 

 say, one of the zeros of life is an ideal point 2£ years before the 

 time of birth. In the formula for the period of maturity or 

 manhood, the place of the ideal zero of vital force is at the age 

 102 years from birth-time nearly. 



When the places of the two ideal zeros of life have been de- 

 termined (say at — 2£ years and at -4- 102 years from birth), 

 there remains only to be determined the point of meeting of the 

 two periods, and the rate of mortality or decrement of life com- 

 mon to both periods at the point of junction. According to the 



